The Fractional Fourier Transform (FrFT) has a wide range of applications in fields such as optics, quantum mechanics, image processing, and communications. The FrFT of a function f(x) of order a is defined asFa[f(x)]=∫−∞∞Ba(x,x′)f(x′)dx′  (1)
where the kernel Ba(x, x′) is defined as
                                          B            a                    ⁡                      (                          x              ,                              x                ′                                      )                          =                                            e                              i                ⁡                                  (                                                            π                      ⁢                                              ϕ                        ^                                            ⁢                                              /                                            ⁢                      4                                        -                                          ϕ                      ⁢                                              /                                            ⁢                      2                                                        )                                                                                                                      sin                  ⁡                                      (                    ϕ                    )                                                                                              1                /                2                                              ×                      e                          i              ⁢                                                          ⁢                              π                (                                                                            x                      2                                        ⁢                                          cot                      ⁡                                              (                        ϕ                        )                                                                              -                                      2                    ⁢                                          xx                      ′                                        ⁢                                          csc                      ⁡                                              (                        ϕ                        )                                                                              +                                                            x                                              ′                        2                                                              ⁢                                          cot                      ⁡                                              (                        ϕ                        )                                                                                            )                                                                        (        2        )            
where φ=ax/2 and {circumflex over (φ)}=sgn[ sin(φ)]. This applies to the range 0<|φ|<π, or 0<|a|<2. In discrete time, the N×1 FrFT of an N×1 vector can be modeled asXa=Fax   (3)
where Fa is an N×N matrix whose elements are given by
                                          F            a                    ⁡                      [                          m              ,              n                        ]                          =                              ∑                                          k                =                0                            ,                              k                ≠                                  (                                      N                    -                    1                    +                                                                  (                        N                        )                                                                    2                        )                                                                              )                                                      N                    ⁢                                                    u                k                            ⁡                              [                m                ]                                      ⁢                          e                                                -                  j                                ⁢                                  π                  2                                ⁢                ka                                      ⁢                                          u                k                            ⁡                              [                n                ]                                                                        (        4        )            
and where uk[m] and uk[n] are the eigenvectors of the matrix S defined by
                    S        =                  (                                                                      C                  0                                                            1                                            0                                            ⋯                                            1                                                                    1                                                              C                  1                                                            1                                            ⋯                                            0                                                                    0                                            1                                                              C                  2                                                            ⋯                                            0                                                                    ⋮                                            ⋮                                            ⋮                                            ⋱                                            ⋮                                                                    1                                            0                                            0                                            ⋯                                                              C                                      N                    -                    1                                                                                )                                    (        5        )                        and                                                                C          n                =                              2            ⁢                          cos              ⁡                              (                                                                            2                      ⁢                      π                                        N                                    ⁢                  n                                )                                              -          4                                    (        6        )            
The FrFT is a useful approach for separating a signal-of-interest (SOI) from interference and/or noise when the statistics of either are non-stationary (i.e., at least one device is moving, Doppler shift occurs, time-varying signals exist, there are drifting frequencies, etc.). The FrFT enables translation of the received signal to an axis in the time-frequency plane where the SOI and interference/noise are not separable in the frequency domain, as produced by a conventional Fast Fourier Transform (FFT), or in the time domain.
The Wigner Distribution (WD) is a time-frequency representation of a signal. The WD may be viewed as a generalization of the Fourier Transform, which is solely the frequency representation. The WD of a signal x(t) can be written asWx(t,f)=∫−∞∞x(t+τ/2)x*(t−τ/2)e−2πjτfdτ  (7)
The projection of the WD of a signal x(t) onto an axis ta gives the energy of the signal in the FrFT domain a, |Xa(t) |2. Letting α=aπ/2, this may be written as|Xα(t)|2=∫−∞∞Wx(t cos(α)−f sin(α), t sin(α)+f cos(α))df   (8)
In discrete time, the WD of a signal x[n] can be written as
                                          W            x                    ⁡                      [                                          n                                  2                  ⁢                                                                          ⁢                                      f                    s                                                              ,                                                kf                  s                                                  2                  ⁢                                                                          ⁢                  N                                                      ]                          =                              e                          j              ⁢                              π                N                            ⁢              kn                                ⁢                                    ∑                              m                =                                  l                  1                                                            l                2                                      ⁢                                                  ⁢                                          x                ⁡                                  [                  m                  ]                                            ⁢                                                x                  *                                ⁡                                  [                                      n                    -                    m                                    ]                                            ⁢                              e                                  j                  ⁢                                      π                    N                                    ⁢                                                                          ⁢                  km                                                                                        (        9        )            
where l1=max (0,n−(N−1)) and l2=min(n,N−1). This particular implementation of the discrete WD is valid for non-periodic signals. Aliasing is avoided by oversampling the signal x[n] using a sampling rate fs (samples per second) that is at least twice the Nyquist rate.
When applying the FrFT to perform interference suppression, the rotational parameter a should first be estimated. Conventional approaches rely on choosing the value of a, 0≦a≦2, which produces a minimum mean-square error (MMSE) between a desired (i.e., training) signal and its estimate. When the environment is non-stationary, it is necessary to perform this estimation with very few samples, i.e., before the statistics of the received signal change significantly. When this is not done, large estimation errors, which result in poor interference suppression, can occur. MMSE-based algorithms, however, require a large number of samples in practice. Hence, their performance will be suboptimal in non-stationary environments. Accordingly, an improved approach to signal separation may be beneficial.